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Figure 4 Appendix: Performance Statistics For Subperiods This table shows summary statistics for different equity indexes. The construction principles of the indexes are more fully described in the text. The statistics are based on weekly data. All statistics are annualized, and performance ratios that involve the average returns are based on the geometric average, which reliably reflects multiple holding period returns for investors. A Cornish-Fisher expansion was used to compute a valueat-risk estimate that takes into account the mean, volatility, skewness and excess kurtosis of index returns. Efficient Index Minimum Volatility Sources: Data from Thomson Reuters and Bloomberg; EDHEC-Risk Institute computations. Fundamental Index S&P 500 Equal-Weighted S&P 500 Russell 1000 Average (geometric) 7.8% 1.7% 6.5% 7.3% -0.1% 0.2% Std. Dev. 16.5% 15.2% 17.2% 18.7% 19.3% 19.4% Semi-deviation (below zero) 11.7% 11.1% 12.1% 13.2% 13.8% 13.9% Tracking error 7.3% 8.0% 7.3% 7.5% 0.0% 1.6% Beta with S&P 500 0.79 0.73 0.83 0.90 1.00 1.00 Sharpe ratio 0.28 -0.10 0.19 0.22 -0.17 -0.15 Sortino ratio 0.67 0.15 0.54 0.55 -0.01 0.02 Information ratio 1.09 0.23 0.91 0.99 NA 0.24 Treynor ratio 0.06 -0.02 0.04 0.05 -0.03 -0.03 95% Value-at-Risk 3.9% 3.6% 4.0% 4.4% 4.5% 4.5% 99% Value-at-Risk 7.2% 6.7% 7.6% 7.8% 7.9% 8.0% Skewness -0.69 -0.50 -0.49 -0.54 -0.34 -0.35 Kurtosis 5.49 5.49 5.74 4.98 4.92 4.98 Efficient Index Noncap-Weighted Minimum Volatility First subperiod (from 1/8/99-7/2/04) Second subperiod (from 7/9/04-1/1/10) Noncap-Weighted Fundamental Index S&P 500 Equal-Weighted Cap-Weighted Cap-Weighted S&P 500 Russell 1000 Average (geometric) 5.0% 3.3% 4.0% 4.2% 1.9% 2.3% Std. Dev. 21.9% 17.9% 23.4% 23.8% 20.3% 20.6% Semi-deviation (below zero) 15.7% 13.2% 16.6% 16.9% 14.7% 14.9% Tracking error 3.7% 4.9% 5.7% 5.2% 0.0% 0.9% Beta with S&P 500 1.07 0.86 1.13 1.16 1.00 1.01 Sharpe ratio 0.10 0.03 0.06 0.06 -0.04 -0.02 Sortino ratio 0.32 0.25 0.24 0.25 0.13 0.16 Information ratio 0.83 0.28 0.38 0.44 NA 0.48 Treynor ratio 0.02 0.01 0.01 0.01 -0.01 0.00 95% VaR (Cornish Fisher) 4.9% 4.1% 5.0% 5.2% 4.6% 4.6% 99% VaR (Cornish Fisher) 12.7% 12.6% 13.7% 13.0% 12.7% 12.8% Skewness -0.55 -1.02 -0.33 -0.40 -0.64 -0.61 Kurtosis 9.81 13.33 10.41 9.14 11.10 10.99 tor models. The R-squared for these regressions indicates how well the factor model fits the observed returns of the respective index, with an R-squared of 1 indicating a perfect fit, meaning that the index would be perfectly replicable with the equity factors. For the cap-weighted indexes, the results in Figure 3 show that the factor model suffices to get an R-squared close to 1. This is not the case for the noncapweighted indexes. They contain some return variation that cannot be explained by the factors. The R-squared reaches levels on the order of 0.95, leaving 5 percent of the return variation unexplained by exposure to common factors. This unexplained part can be linked to the unique index construction method of each of the noncap-weighted indexes. Conclusion The analysis provided in this paper clearly shows that the noncap-weighted indexes beat the standard cap-weighted indexes such as the S&P 500 and the Russell 1000 in terms of www.journalofindexes.com January / February 2011 17
Figure 5 Appendix: Risk/Reward Difference Compared To S&P 500 For Subperiods This table shows differences in average returns, in volatility and in Sharpe ratios between each index and the cap-weighted S&P 500 and the associated p-values. All statistics are computed from weekly returns data. All differences are computed from annualized statistics and the average returns are a geometric average. The p-values for differences are computed using a paired t-test for the average returns, a Fisher test for the volatility and a Jobson-Korkie test for the Sharpe ratio. Differences that are significantly different from zero at the 5% level are indicated in bold. Noncap-Weighted Efficient Index Minimum Volatility Fundamental Index S&P 500 Equal-Weighted Diff. in average 8.0% 1.8% 6.6% 7.4% p-value 2.1% 74.2% 5.1% 2.7% Diff. in volatility -2.8% -4.1% -2.1% -0.6% p-value 0.8% 0.0% 5.3% 62.4% Diff. in Sharpe ratio 0.45 0.07 0.37 0.39 p-value 0.9% 75.1% 3.6% 2.5% Sources: Data from Thomson Reuters and Bloomberg; EDHEC-Risk Institute computations. First subperiod (from 1/8/99-7/2/04) Second subperiod (from 7/9/04-1/1/10) Noncap-Weighted Efficient Index Minimum Volatility Fundamental Index S&P 500 Equal-Weighted Diff. in average 3.1% 1.4% 2.1% 2.3% p-value 3.6% 67.3% 25.3% 18.0% Diff. in volatility 1.7% -2.4% 3.2% 3.5% p-value 18.2% 3.6% 1.4% 0.7% Diff. in Sharpe ratio 0.14 0.07 0.10 0.10 p-value 2.9% 54.7% 23.5% 12.2% risk-adjusted performance over the 11 years for which data is available for all indexes. Moreover, the “improved beta” strategies achieve the main objectives—which vary widely from one noncapweighted index to another—that they have set for themselves. The efficient index, whose aim is higher risk/reward efficiency, does indeed obtain the highest Sharpe ratio of all the indexes. The minimum-volatility index obtains the lowest volatility. Equal-weighted indexes and fundamental indexes obtain higher average returns as a result of their mechanical rebalancing feature. The differences among the indexes are also reflected in quite different factor exposures. Minimum volatility, as it favors low beta stocks, clearly has the lowest market beta of all noncap-weighted indexes, while fundamental weighting leads to the highest value exposure, and equal weighting to the most pronounced anti-momentum exposure. In sum, efficient indexes, the newest of the strategies analyzed in this paper, not only obtain the highest Sharpe ratios in the period analyzed in this paper, but also appear to have the most diversified factor exposures. Their volatility is lower than that of equal-weighted and fundamental-weighted indexes, and since they are not subject to the same tilt toward low beta stocks, their average returns are higher than those of minimum-volatility indexes. Efficient indexes, which are an attempt to provide an investable proxy for the tangency portfolio from modern portfolio theory, are thus a significant new addition to the improved beta toolbox. That the four weighting schemes have rather different risk and return properties also suggests that different investors may choose different alternatives, depending on which characteristic they value most. Moreover, combining these alternatives may add even more value, as different return properties may allow an investor who wants to move away from capitalization weighting to diversify his exposure to noncap-weighted schemes. This issue is left for further research. Appendix Summary Statistics And Differences With Capitalization Weighting In Subperiods Figures 4 and 5 contain an analysis of results for the two subperiods of the sample. On the whole, the results per subperiod confirm the conclusions drawn for the full period. The results for the full period can thus be considered quite robust. Additional Description Of Data Sources For the noncap-weighted indexes, the data for MSCI and S&P EW is obtained from Datastream. The information on 18 January / February 2011
Page 1 and 2: Improved Beta? Noël Amenc, Felix G
Page 3 and 4: Contributors Noël Amenc Noël Amen
Page 5 and 6: FREE SUBSCRIPTION OFFER! The Journa
Page 7 and 8: David Smith Publisher dsmith@fa-mag
Page 9 and 10: Editor’s Note Considering Alterna
Page 11 and 12: Improved Beta? A comparison of inde
Page 13 and 14: volatility and correlation (for min
Page 15 and 16: Figure 2 Risk/Reward Difference Com
Page 17: show marginal statistical significa
Page 21 and 22: The Evolution Of Equal Weighting Ad
Page 23 and 24: Figure 3 1 0.8 0.6 0.4 0.2 0 Figure
Page 25 and 26: Russell’s sector equal-weighted i
Page 27 and 28: structures in the U.S. large- and s
Page 29 and 30: negative alpha is observed for both
Page 31 and 32: The Weight Debate A roundtable 30 J
Page 33 and 34: for perceived growth is constantly
Page 35 and 36: isks and active risks in active str
Page 37 and 38: Core Vs. Blend Index investing in t
Page 39 and 40: that given size range. In contrast,
Page 41 and 42: Benchmarking Microfinance Equity In
Page 43 and 44: Figure 1 Illustration: Determining
Page 45 and 46: Figure 3 Establishing MFI SVIX Annu
Page 47 and 48: Figure 5 WSAS MFI SVIX Series - Bas
Page 49 and 50: aggregation and resultant “smooth
Page 51 and 52: Talking Indexes Sorting And Digging
Page 53 and 54: News Indexes Regain Edge Over Activ
Page 55: News version of it, which excludes
Page 58 and 59: policies won’t change, Guggenheim
Page 60 and 61: Global Index Data Selected Major In
Page 62 and 63: Morningstar U.S. Style Overview Jan
Page 64 and 65: Exchange-Traded Funds Corner Larges
Page 66 and 67: INVESTING IN THE FUTURE OF THE U.S.
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